Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,4,8,6)(9,10,13,11)(12,15,16,14)(19,25)(20,27)(23,28)(24,29)(34,38)(35,41)(40,45)(43,47);(1,3,5,8)(2,6,7,4)(9,11,13,10)(12,15,16,14)(18,22)(23,28)(24,29)(30,32)(36,44)(40,45)(43,47)(46,48);(1,2,5,7)(3,4,8,6)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2)(3,4)(5,7)(6,8)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4)(2,3)(5,6)(7,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fghi,c^2=gh,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5030830);